Systems view of sampling and reconstruction 1.20 2003/07/18 02:40:00 GMT-5 2004/03/02 05:07:03.863 US/Central Anders Gjendemsjo gjendems@NO-SPAM.tele.ntnu.no Anders Gjendemsjo gjendems@NO-SPAM.tele.ntnu.no Sampling Reconstruction Hold System An overview of sampling and reconstruction on a system level.

Ideal reconstruction system shows the ideal reconstruction system based on the results of the Sampling theorem proof. consists of a sampling device which produces a time-discrete sequence xsn. The reconstruction filter, ht, is an ideal analog sinc filter, with h t sinc t Ts . We can't apply the time-discrete sequence xsn directly to the analog filter ht. To solve this problem we turn the sequence into an analog signal using delta functions. Thus we write xs t n xs n δ t n T .

Ideal reconstruction system

But when will the system produce an output x̂ t x t ? According to the sampling theorem we have x̂ t x t when the sampling frequency, Fs, is at least twice the highest frequency component of xt.

Ideal system including anti-aliasing To be sure that the reconstructed signal is free of aliasing it is customary to apply a lowpass filter, an anti-aliasing filter, before sampling as shown in .

Ideal reconstruction system with anti-aliasing filter

Again we ask the question of when the system will produce an output x̂ t s t ? If the signal is entirely confined within the passband of the lowpass filter we will get perfect reconstruction if Fs is high enough. But if the anti-aliasing filter removes the "higher" frequencies, (which in fact is the job of the anti-aliasing filter), we will never be able to exactly reconstruct the original signal, st. If we sample fast enough we can reconstruct xt, which in most cases is satisfying. The reconstructed signal, x̂ t , will not have aliased frequencies. This is essential for further use of the signal.

Reconstruction with hold operation To make our reconstruction system realizable there are many things to look into. Among them are the fact that any practical reconstruction system must input finite length pulses into the reconstruction filter. This can be accomplished by the hold operation. To alleviate the distortion caused by the hold opeator we apply the output from the hold device to a compensator. The compensation can be as accurate as we wish, this is cost and application consideration.

More practical reconstruction system with a hold component

By the use of the hold component the reconstruction will not be exact, but as mentioned above we can get as close as we want.

Introduction Proof Illustrations Matlab example Hold operation Aliasing applet Exercises